The determinant of one-dimensional polyharmonic operators of arbitrary order

Abstract

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_n=(-1{)}^n({\partial }_x{)}^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies the asymptotics $\log (\det P_n)=-n^2\log n+\left[\frac{7\zeta (3)}{2{\pi }^2}+\frac{3}{2}+\log \left(\frac{T}{4}\right)\right]n^2+\mathrm{O}(n)$ for large $n$. This is a consequence of sharp upper and lower bounds for $\log (\det P_n)$ valid for all $n$ and which coincide in the terms up to order $n$. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.

This version further extends results to a large class of general polyharmonic operators with variable coefficients. 28 pages, 1 figure